Planar Harmonic Mappings With a Given Jacobian

The article is devoted to the study of the Jacobians of sense-preserving harmonic mappings in the unit disk of the complex plane. The main result is a criterion for an infinitely differentiable positive function to be a Jacobian of some sense-preserving harmonic mapping. The relationship between a J...

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Bibliographic Details
Published in:	Issues of analysis Vol. 30; no. 3; pp. 69 - 85
Main Authors:	Graf, S. Yu, Nikitin, I. A.
Format:	Journal Article
Language:	English
Published:	Petrozavodsk State University 01-01-2023
Subjects:	dilatation jacobian planar harmonic mappings schwarzian derivative
Online Access:	Get full text
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Summary:	The article is devoted to the study of the Jacobians of sense-preserving harmonic mappings in the unit disk of the complex plane. The main result is a criterion for an infinitely differentiable positive function to be a Jacobian of some sense-preserving harmonic mapping. The relationship between a Jacobian of a harmonic mapping and the Schwarzian derivative of its dilatation is revealed. The structure of the set of harmonic mappings with a given Jacobian is described. The results are illustrated by examples. In conclusion, we consider an application of the main results of the article to the construction of variational formulas in classes of harmonic mappings with a given Jacobian.
ISSN:	2306-3432 2306-3424 2306-3432
DOI:	10.15393/j3.art.2023.14110